scholarly journals The orthogonality of Hecke eigenvalues

2007 ◽  
Vol 143 (03) ◽  
pp. 541-565 ◽  
Author(s):  
Henryk Iwaniec ◽  
Xiaoqing Li
Keyword(s):  
2019 ◽  
Vol 198 ◽  
pp. 139-158 ◽  
Author(s):  
Ping Song ◽  
Wenguang Zhai ◽  
Deyu Zhang

2018 ◽  
Vol 2020 (19) ◽  
pp. 6149-6168
Author(s):  
Michael Lipnowski ◽  
George J Schaeffer

Abstract We describe a novel method for bounding the dimension $d$ of the largest simple Hecke submodule of $S_{2}(\Gamma _{0}(N);\mathbb{Q})$ from below. Such bounds are of interest because of their relevance to the structure of $J_{0}(N)$, for instance. In contrast with previous results of this kind, our bound does not rely on the equidistribution of Hecke eigenvalues. Instead, it is obtained via a Hecke-compatible congruence between the target space and a space of modular forms whose Hecke eigenvalues are easily controlled. We prove conditional bounds, the strongest of which is $d\gg _{\epsilon } N^{1/2-\epsilon }$ over a large set of primes $N$, contingent on Soundararajan’s heuristics for the class number problem and Artin’s conjecture on primitive roots. For prime levels $N\equiv 7\mod 8,$ our method yields an unconditional bound of $d\geq \log _{2}\log _{2}(\frac{N}{8})$, which is larger than the known bound of $d\gg \sqrt{\log \log N}$ due to Murty–Sinha and Royer. A stronger unconditional bound of $d\gg \log N$ can be obtained in more specialized (but infinitely many) cases. We also propose a number of Maeda-style conjectures based on our data, and we outline a possible congruence-based approach toward the conjectural Hecke simplicity of $S_{k}(\textrm{SL}_{2}(\mathbb{Z});\mathbb{Q})$.


2019 ◽  
Vol 159 (1) ◽  
pp. 287-298
Author(s):  
H. F. Liu ◽  
R. Zhang
Keyword(s):  

2014 ◽  
Vol 89 (4) ◽  
pp. 979-1014 ◽  
Author(s):  
Étienne Fouvry ◽  
Satadal Ganguly ◽  
Emmanuel Kowalski ◽  
Philippe Michel

2019 ◽  
Vol 31 (2) ◽  
pp. 403-417
Author(s):  
Youness Lamzouri

AbstractLet f be a Hecke cusp form of weight k for the full modular group, and let {\{\lambda_{f}(n)\}_{n\geq 1}} be the sequence of its normalized Fourier coefficients. Motivated by the problem of the first sign change of {\lambda_{f}(n)}, we investigate the range of x (in terms of k) for which there are cancellations in the sum {S_{f}(x)=\sum_{n\leq x}\lambda_{f}(n)}. We first show that {S_{f}(x)=o(x\log x)} implies that {\lambda_{f}(n)<0} for some {n\leq x}. We also prove that {S_{f}(x)=o(x\log x)} in the range {\log x/\log\log k\to\infty} assuming the Riemann hypothesis for {L(s,f)}, and furthermore that this range is best possible unconditionally. More precisely, we establish the existence of many Hecke cusp forms f of large weight k, for which {S_{f}(x)\gg_{A}x\log x}, when {x=(\log k)^{A}}. Our results are {\mathrm{GL}_{2}} analogues of work of Granville and Soundararajan for character sums, and could also be generalized to other families of automorphic forms.


2020 ◽  
Vol 16 (06) ◽  
pp. 1185-1197
Author(s):  
Chi-Yun Hsu

Let [Formula: see text] be a modular form with complex multiplication. If [Formula: see text] has critical slope, then Coleman’s classicality theorem implies that there is a [Formula: see text]-adic overconvergent generalized Hecke eigenform with the same Hecke eigenvalues as [Formula: see text]. We give a formula for the Fourier coefficients of this generalized Hecke eigenform. We also investigate the dimension of the generalized Hecke eigenspace of [Formula: see text]-adic overconvergent forms containing [Formula: see text].


2019 ◽  
Vol 141 (2) ◽  
pp. 485-501 ◽  
Author(s):  
Wenzhi Luo ◽  
Fan Zhou

2017 ◽  
Vol 13 (05) ◽  
pp. 1213-1231
Author(s):  
Avner Ash ◽  
David Pollack

Starting with a numerically noncritical (at [Formula: see text]) Hecke eigenclass [Formula: see text] in the homology of a congruence subgroup [Formula: see text] of [Formula: see text] (where [Formula: see text] divides the level of [Formula: see text]) with classical coefficients, we first show how to compute to any desired degree of accuracy a lift of [Formula: see text] to a Hecke eigenclass [Formula: see text] with coefficients in a module of [Formula: see text]-adic distributions. Then we show how to find to any desired degree of accuracy the germ of the projection [Formula: see text] to weight space of the eigencurve around the point [Formula: see text] corresponding to the system of Hecke eigenvalues of [Formula: see text]. We do this under the conjecturally mild hypothesis that [Formula: see text] is smooth at [Formula: see text].


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